Systems and methods for model-based estimation of cardiac output and total peripheral resistance

ABSTRACT

The methods and systems for estimating cardiac output and total peripheral resistance include observing arterial blood pressure waveforms to determine intra-beat and inter-beat variability in arterial blood pressure and estimating from the variability a time constant for a lumped parameter beat-to-beat averaged Windkessel model of the arterial tree. Uncalibrated cardiac output and uncalibrated total peripheral resistance may then be calculated from the time constant. Calibrated cardiac output and calibrated total peripheral resistance may be computed using calibration data, assuming an arterial compliance that is either constant or dependent on mean arterial blood pressure. The parameters of the arterial compliance may be estimated in a least-squares manner.

CROSS REFERENCE TO RELATED APPLICATIONS

The instant application claims priority from provisional application No.60/938,253 filed May 16, 2007, the disclosure of which is incorporatedherein by reference in its entirety.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

This invention was made with Government support under Contract No. R01EB001659, awarded by the National Institutes of Biomedical Imaging andBioengineering (NIBIB), a part of the National Institute of Health(NIH), and Grant No. CA00403, awarded by the National Space BiomedicalResearch Institute (NSBRI). The Government has certain rights in thisinvention.

FIELD OF THE INVENTION

This invention relates to cardiac output and total peripheral resistanceestimation, and more particularly to cardiac output and total peripheralresistance estimation from peripheral or central arterial blood pressurewaveforms.

BACKGROUND OF THE INVENTION

Cardiac output (CO) is the amount of blood the heart pumps out over aunit of time. Typical values of CO in resting adults range from 3liters/minute to 6 liters/minute. One basis for estimating or measuringCO is the formula CO=HR×SV, where SV is cardiac stroke volume and HR isheart rate. If SV is measured in liters/beat and HR is measured in beatsper minute, then CO is given in liters/minute, although any other unitsof volume and time may be used. Another basis for estimating ormeasuring CO is the formula CO=MAP/TPR, where MAP is mean arterial bloodpressure and TPR is total peripheral resistance.

Cardiac output (CO) is a key hemodynamic variable that is commonly usedto establish differential diagnoses, monitor disease progression, andtitrate therapy in many cardiovascular conditions. For example, whencombined with estimates of other hemodynamic variables such as meanarterial blood pressure (MAP) and total peripheral resistance (TPR),estimates of cardiac output may allow clinicians to determine the causeof circulatory shock [1]. (Numbers in square brackets refer to thereference list included herein. The contents of all these references areincorporated herein by reference.)

The current clinical gold-standard for measuring CO is intermittentthermodilution, a highly invasive procedure in which a balloon-tippedcatheter (Swan-Ganz catheter [2]) is advanced to the pulmonary artery, abolus of cold saline is injected into the circulation, and the blood'stemperature profile is observed as a function of time. Due to its highdegree of invasiveness, this procedure is usually reserved for only thesickest of patients, and even in critical care its benefit isincreasingly questioned as retrospective clinical studies in the pastten years conclude that the use of a pulmonary artery catheter may notimprove patient outcome [3], [4]. There are several patents thatdisclose systems directed to estimating CO via thermodilution. Someexamples include U.S. Pat. No. 4,236,527 to Newbower et al., U.S. Pat.No. 4,507,974 to Yelderman, U.S. Pat. No. 5,146,414 to McKown et al.,and U.S. Pat. No. 5,687,733 to McKown et al. The disadvantage of thesesystems is that they are highly invasive, and that CO can only bemeasured intermittently. In many situations, e.g. in critical careunits, CO measurements via thermodilution may be made only every 2-3days.

It is possible, however, to obtain estimates of cardiac output withoutusing highly invasive procedures: rather than intermittently measuringaverage cardiac output invasively via thermodilution, many attempts havebeen made to estimate CO from the arterial blood pressure (ABP) waveform[5], [6], [7], [8], [9], [10], [11], using models of the arterialsystem. One of the most basic of these models is the Windkessel model[5] (see FIG. 3 a), in which the arterial tree is modeled as a single,leaky pressurized chamber that is filled intermittently with boluses offluid. Because HR is usually easy to measure using any of a wide varietyof instruments, the calculation of CO usually depends on some techniquefor estimating stroke volume. Conversely, any method that yields a valuefor CO can be used to determine SV. In addition, estimates of CO (or SV)can be used with HR to estimate any parameter that can be derived fromeither of these values.

An entire class of patented or patent-pending algorithms is based onanalyzing the pressure pulse morphology, often in the context ofWindkessel-like models for the arterial tree [6], [7], [8], [9], [12].Examples of these are U.S. Pat. No. 5,400,793 to Wesseling, U.S. PatentApplication Publication No. 20050124903 to Roteliuk et al., U.S. PatentApplication Publication No. 20050124904 to Roteliuk, U.S. PatentApplication Publication No. 20060235323 to Hatib et al., U.S. PatentApplication Publication No. 20080015451 to Hatib et al., the contents ofeach of which are incorporated herein in their entirety. In many ofthese, since stroke volume is related to the arterial pressure pulsethrough the properties of the arterial tree, SV (and hence CO) isestimated on an intra-cycle timescale using morphological features ofeach individual ABP wavelet (such as systolic, mean, or diastolic ABP).One significant disadvantage of most of these methods or systems forestimating cardiac output is that they do not provide beat-to-beatestimates of cardiac output.

More recently, Cohen et al. ([10], [13], and U.S. Patent ApplicationPublication No. 20040158163 to Cohen et al., the contents of which areincorporated herein in their entirety) intermittently, i.e. every 3minutes, estimated relative changes in cardiac output from theinter-cycle (or beat-to-beat) variations of the ABP waveform, usingthese to determine the impulse response function of a model ofsignificantly higher order than the Windkessel model and, from it, thetime constant of arterial outflow that would be associated with aWindkessel model. Knowing the latter, the authors determinedproportional CO, from which absolute CO can be obtained via calibrationwith a single or multiple reference CO measurements. In theircalibration, Cohen et al. assume a linear relationship of arterialvolume to mean pressure relationship, corresponding to constant lumpedarterial tree compliance in the Windkessel model. Applicants' owninterest in estimating CO and TPR derives from their own work in thearea of cycle-averaged models of the cardiovascular system [14], [15],[16], where again the focus was on inter-cycle variation.

A criticism of Cohen et al. put forward in U.S. Patent ApplicationPublication No. 20060235323 to Hatib et al. is that the approachdisclosed by Cohen requires determination of a calibration factor onwhich accuracy of the CO measurement is closely dependent. Hatib et al.argue that Cohen's method ignores much of the information contained inthe pressure waveform. In fact, as Hatib et al. note, one embodiment ofCohen's method uses only a single characteristic of each waveform,namely the area. Hatib et al. also note that a partial consequence ofCohen's greatly-simplified input signal to his recursive model is theneed for a complicated transfer function model, which involves manyzeroes, many poles, and, consequently, design and computationalcomplexity.

However, Applicants note that the cardiac output estimation apparatusand methods described in U.S. Patent Application Publication No.20050124903 to Roteliuk et al., U.S. Patent Application Publication No.20050124904 to Roteliuk, U.S. Patent Application Publication No.20060235323 to Hatib et al., U.S. Patent Application Publication No.20080015451 to Hatib et al. (commonly owned by Edwards Life SciencesCorporation, hereinafter “Edwards”) and in Cohen et al. explicitlyassume an impulsive input flow waveform. Furthermore, the methods ofEdwards and Cohen require a fixed sampling rate, i.e., the rate at whichthe impulsive input flow waveform is generated and/or sampled. There isa need for CO and TPR estimation methods that do not require theassumption of such an input waveform, and that do not require fixedsampling rates. The methods of Cohen and Edwards also explicitly useactual arterial blood pressure waveforms, which make them moresusceptible to noise and artifacts inherent in these waveforms. There isa need for CO and TPR estimation methods that use parameters orvariables derived from blood pressure waveforms instead of bloodpressure waveforms that are sampled at a very high rate, e.g., 90 Hz orgreater within each cycle.

The Edwards patents, collectively, and U.S. Pat. No. 5,400,793 toWesseling (hereinafter “Wesseling”) assume a 3-element Windkessel modelin which a value for the input impedance of the model is either assumedor estimated. As the number of elements in a model increases, so doesthe complexity of the processing tasks that must be carried out toestimate CO or TPR. Therefore, the parameters and variables in thismodel cannot be easily estimated without making several assumptions, andrequiring more input data than may be available in settings such ascritical care units. The Edwards patents and Wesseling also describecalibration schemes for calibrating uncalibrated cardiac output, i.e.,for calculating a multiplicative calibration factor that is used toobtain absolute cardiac output from proportional or relative cardiacoutput. In Edwards and Wesseling, the calibration scheme is dependent oncoarse patient-specific data, e.g., height, body mass, age, gender.Wesseling's calibration factor has 3 parameters. In Edwards, thecalibration factor furthermore requires the calculation of moments ofthe arterial blood pressure waveform. The calibration factors describedin Edwards and Wesseling are complicated functions of three or moreparameters which require several (at times, patient-specific) inputs.The Wesseling calibration factor is only grossly correlated to thecardiovascular system model he describes. There is still a need forsimpler calibration factors that require fewer inputs and/orpatient-specific parameters.

Although many CO estimation methods exist, as described above, there isstill a need for CO estimation algorithms that are robust, and thateffectively exploit both inter-cycle and intra-cycle variations in theblood pressure waveform. Thus, there is a need for CO and TPR estimationmethods that use parameters or variables derived from blood pressurewaveforms instead of highly-sampled blood pressure waveforms themselves.There also exists a need for CO estimation algorithms in which relativecardiac output estimates can be easily calibrated to obtain absolutecardiac output estimates. Thus, there is still a need for simplercalibration factors that require fewer inputs and/or patient-specificparameters. Current cardiac output estimation algorithms are not robustin the sense that they may perform well on a particular data set, butpoorly on a different data set. There have been many methods in the pastthat seemed promising, but turned out not to work robustly. Furthermore,these CO estimation algorithms generally exploit either inter-cycle orintra-cycle variability. Currently, there are no algorithms forestimating cardiac output or total peripheral resistance thateffectively exploit both inter-cycle and intra-cycle variations in theABP waveform to estimate CO or TPR.

SUMMARY OF THE INVENTION

According to one aspect, the invention relates to a method forestimating beat-by-beat cardiovascular parameters and variables,comprising processing one or more cycles of arterial blood pressure todetermine intra-beat and inter-beat variability in blood pressure, andcomputing estimates of one or more cardiovascular parameters andvariables from the intra-beat variability, the inter-beat variability,and a beat-to-beat averaged Windkessel model of an arterial tree.

In an embodiment, the cardiovascular system parameters include abeat-by-beat time constant of the arterial tree. In an embodiment, thetime constant is estimated over a data window. Optionally, the timeconstant is estimated through optimization of an error criterion. Thiserror criterion may be a least-squared error criterion.

In certain embodiments, the cardiovascular system variables include anuncalibrated beat-by-beat cardiac output. In some embodiments, themethod further comprises computing calibrated beat-by-beat cardiacoutput from the uncalibrated beat-by-beat cardiac output using acalibration factor. In an embodiment, the calibration factor is computedfor each of the cycles.

In an embodiment, the calibration factor represents a lumped arterialcompliance. This lumped arterial compliance may be modeled as a functionof mean arterial blood pressure. Optionally, this lumped arterialcompliance is modeled as a parameterized function of mean arterial bloodpressure. Alternatively, this lumped arterial compliance is modeled as atwo-parameter function of mean arterial blood pressure, or as aconstant. The parameters of this lumped arterial compliance may beestimated through optimization of an error criterion. In someembodiments, the criterion is a least-squared error.

In another embodiment, the cardiovascular parameters includeuncalibrated beat-by-beat total peripheral resistance. In someembodiments, the method further comprises computing calibratedbeat-by-beat total peripheral resistance from the ratio of the timeconstant to a lumped arterial compliance. In an embodiment, the methodfurther comprises computing calibrated beat-by-beat total peripheralresistance from the ratio of mean arterial blood pressure to calibratedcardiac output. In another embodiment, the method further comprisescomputing calibrated beat-by-beat total peripheral resistance from theratio of mean pressure to a systemic blood flow. In some embodiments,the systemic blood flow is calibrated cardiac output minus the productof lumped arterial compliance with the ratio of beat-to-beat arterialblood pressure change to beat duration.

In some embodiments, the arterial blood pressure is measured at acentral artery of the cardiovascular system. In other embodiments, thearterial blood pressure is measured at a peripheral artery of thecardiovascular system. In certain embodiments, the arterial bloodpressure is measured using a noninvasive blood pressure device. Thisdevice may be a photoplethysmographic or tonometric blood pressuredevice.

In a further embodiment, processing the one or more cycles of arterialblood pressure includes obtaining values for mean blood pressure,diastolic blood pressure, and systolic blood pressure for each cycle. Insome embodiments, processing the one or more cycles of arterial bloodpressure includes obtaining an onset time for each cycle. In a furtherembodiment, processing the one or more cycles of arterial blood pressureincludes computing a beat-to-beat arterial blood pressure change betweenconsecutive onset times. Optionally, processing the one or more cyclesof arterial blood pressure includes estimating pulse pressure in eachcycle as a proportionality constant multiplied by a difference betweenmean pressure and diastolic pressure in each cycle. In some embodiments,the proportionality constant in each cycle is fixed. In someembodiments, the proportionality constant is greater than 1 and lessthan 4, but preferably 2. In certain embodiments, processing the one ormore cycles of arterial blood pressure includes obtaining a beatduration for each cycle.

In another aspect, the invention relates to a system for estimatingbeat-to-beat cardiac output comprising a blood pressure measuringdevice, a processor, a display, a user interface, and a memory storingcomputer executable instructions, which when executed by the processorcause the processor to receive one or more cycles of arterial bloodpressure from the blood pressure device, analyze one or more cycles ofarterial blood pressure to determine intra-beat and inter-beatvariability in blood pressure, compute estimates of one or morecardiovascular system parameters and variables from the intra-beatvariability, the inter-beat variability, and a beat-to-beat averagedWindkessel model of an arterial tree, and display the estimates. Incertain practice, the blood pressure device is a noninvasive bloodpressure device. Optionally, this device may be a photoplethysmographicor a tonometric blood pressure device.

BRIEF DESCRIPTION OF THE DRAWINGS

The invention may be better understood from the following illustrativedescription with reference to the following drawings:

FIG. 1 is a block diagram of a system for estimating cardiac output andtotal peripheral resistance, according to an illustrative embodiment ofthe invention;

FIG. 2 is a process flow diagram suitable for estimating cardiac outputand total peripheral resistance with the system of FIG. 1, according toan illustrative embodiment of the invention;

FIG. 3 a is a circuit representation for the Windkessel model;

FIG. 3 b is a graph illustrating a representative pulsatile arterialblood pressure waveform for the Windkessel circuit model of FIG. 3 a;

FIG. 4 is a graph of arterial blood pressure versus time showing aplurality of cycles of a porcine radial arterial blood pressurewaveform;

FIG. 5 is a table showing the characteristics of a porcine data setaccording to an animal experiment disclosed herein;

FIG. 6 is a table summarizing the results of an animal experimentdisclosed herein;

FIG. 7 is a table summarizing linear regressions of estimated versustrue cardiac output for animal experiments disclosed herein;

FIG. 8 is a table summarizing linear regressions of estimated cardiacoutput versus heart rate, mean blood pressure, and true cardiac outputfor animal experiments disclosed herein;

FIG. 9 includes graphs of true and estimated cardiac output and true andestimated total peripheral resistance according to a first animalexperiment disclosed herein;

FIG. 10 includes graphs of true and estimated cardiac output and trueand estimated total peripheral resistance according to a second animalexperiment disclosed herein;

FIG. 11 includes graphs of true and estimated cardiac output and trueand estimated total peripheral resistance according to a third animalexperiment disclosed herein;

FIG. 12 is a Bland-Altman plot of cardiac output estimation error versusthe mean of true and estimated cardiac output;

FIG. 13 is a graph of a linear regression of estimated versus truecardiac output;

FIG. 14 is a table summarizing the results of the application of themethod of Cohen et al. described in U.S. Patent Publication 20040158163;

FIG. 15 is a table showing a comparison of Applicants' method forestimating cardiac output and other well-known methods from theliterature;

FIG. 16 is a table showing a comparison of results using Applicants'mean pressure-dependent compliance calibration factor and a constantcompliance calibration factor;

FIG. 17 includes graphs of true and estimated cardiac output and a meanpressure-dependent lumped arterial calibration factor according to afirst animal experiment disclosed herein;

FIG. 18 includes graphs of the ratio of true to estimated cardiac outputversus mean pressure for each of the animal experiments disclosedherein;

FIG. 19 is a table showing a comparison of results obtained usingApplicants' method and the Herd method in which both sets of results areobtained using Applicants' mean pressure-dependent compliancecalibration factor; and

FIG. 20 is a table showing a comparison of results obtained usingApplicants' method and the Herd method in which both sets of results areobtained using a constant compliance calibration factor.

DESCRIPTION OF CERTAIN ILLUSTRATIVE EMBODIMENTS

To provide an overall understanding of the invention, certainillustrative embodiments will now be described. However, it will beunderstood by one of ordinary skill in the art that the methodsdescribed herein may be adapted and modified as is appropriate for theapplication being addressed and that the systems and methods describedherein may be employed in other suitable applications, and that suchother additions and modifications will not depart from the scope hereof.

FIG. 1 is a block diagram of a cardiac output estimation system 100 inwhich the present invention's teachings may be implemented. The cardiacoutput estimation system 100 includes blood pressure measuring device102, processor 104, memory 106 e.g. Read-Only Memory (ROM), display 108,and user interface 110. The processor 104 operates on blood pressuredata in accordance with computer executable instructions loaded intomemory 106. The instructions will ordinarily have been loaded into thememory from local persistent storage in the form of, say, a disc drivewith which the memory communicates. The instructions may additionally orinstead be received by way of user interface 110. The system may alsoreceive user input from devices such as a keyboard, mouse, ortouch-screen. Blood pressure measuring device 102 may be an invasive ora noninvasive device. Blood pressure measuring device 102 may be aphotoplethysmographic or tonometric device. The blood pressure may bemeasured at a central, a pulmonary, or a peripheral artery in thecardiovascular system.

FIG. 2 is a process flow diagram 200 including steps 202, 204, 206, 208,210, 216, and 218, suitable for estimating cardiac output and totalperipheral resistance with the cardiac output estimation system 100 ofFIG. 1, according to an illustrative embodiment of the invention. Bloodpressure measuring device 102 measures arterial blood pressure waveformsand transmits the ABP waveform data to processor 104 of cardiac outputestimation system 100 of FIG. 1. In step 202 of process 200, processor104 of cardiac output estimation system 100 of FIG. 1 receives one ormore cycles of ABP data and processes the ABP data. As will be explainedin detail with reference to FIGS. 3 and 4, the outputs 212 of step 202may include mean arterial blood pressure, diastolic arterial bloodpressure, systolic arterial blood pressure, cycle onset time, and cyclelength, for each of the received cycles of the arterial blood pressure.In step 204, as will be explained below in reference to FIGS. 3-20,processor 104 of cardiac output estimation system 100 estimates the timeconstant for a beat-to-beat averaged Windkessel model. This timeconstant is used in step 206, as explained below in reference to FIGS.3-20, to obtain an uncalibrated estimate of cardiac output or totalperipheral resistance. Calibrated cardiac output or total peripheralresistance may be obtained using true cardiac output measurements 208and a model for a lumped arterial compliance calibration factor, as willbe described below in reference to FIGS. 3-20, in steps 210 and 214.Total peripheral resistance may be estimated, as will be described belowin reference to FIGS. 3-20. Some estimates of TPR may require the use ofestimated calibrated cardiac output in step 218. The results of process200 may include, among other cardiovascular system parameters andvariables, estimated calibrated cardiac output 216 and estimated totalperipheral resistance 220, as will be explained below in reference toFIGS. 3-20.

In this manner, cardiac output and total peripheral resistance may beestimated robustly because cardiac output estimation system 100 of FIG.1 uses intra- and inter-beat variability in the arterial blood pressurewaveforms, instead of the actual waveforms themselves. As will bedescribed below in reference to FIGS. 3-20, the results obtained usingthe cardiac output estimation system 100 of FIG. 1 are much better thanthose obtained with other CO estimation systems or methods in theliterature.

In the discussion below, Applicants will describe some embodiments inmore detail. Applicants will begin with a description of the Windkesselmodel with reference to FIGS. 3 a and 3 b. Although only the two-elementWindkessel model is presented here, the derivations and results caneasily be extended to Windkessel models with 3 or more elements. Thisdescription will be followed by a derivation of the beat-to-beataveraged Windkessel model and a linear least-squares estimation ofparameters and variables of this model, with reference to arterial bloodpressure waveforms in FIG. 4. These waveforms are representative ofthose that may be processed at step 202 of FIG. 2 by cardiac outputestimation system 100 of FIG. 1. Applicants conclude this section with adetailed description of experimental results using an animal (porcine orpig or swine) data set, with reference to FIGS. 5-20.

The Windkessel Model

The Windkessel model describes the basic morphology of an arterialpressure pulse [5]. It lumps the distributed resistive and capacitiveproperties of the entire arterial tree into two elements, as seen in theelectrical circuit analog in FIG. 3 a: a single resistor R, representingtotal peripheral resistance (TPR), and a single capacitor C,representing the aggregate elastic properties of all systemic arteries.

The differential equation representing the Windkessel circuit at time tis given by

$\begin{matrix}{{{C\frac{\mathbb{d}{P(t)}}{\mathbb{d}t}} + \frac{P(t)}{R}} = {Q(t)}} & ( {{EQ}.\mspace{14mu} 1} )\end{matrix}$where P(t) represents arterial blood pressure at the aortic root at timet. In this application, at times Applicants use P and V_(u)interchangeably. This equation shows that the time constant τ=RC governsthe intra-cycle dynamics of the Windkessel model. The same time constantalso governs the inter-cycle dynamics, as noted in [10], [14], [15].

The pumping action of the heart is represented by an impulsive currentsource Q(t) that deposits a stroke volume SV_(n) into the arterialsystem during the n^(th) cardiac cycle:

$\begin{matrix}{{Q(t)} = {\sum\limits_{n}\;{{SV}_{n} \cdot {\delta( {t - t_{n}} )}}}} & ( {{EQ}.\mspace{14mu} 2} )\end{matrix}$where t_(n) is the onset time of the n^(th) beat and δ(t) is the unitDirac impulse. It then follows by integrating (EQ. 1) over just the(infinitesimal) ejection phase thatSV _(n) =C·PP _(n)   (EQ. 3)where PP_(n) is the pulse pressure in the n^(th) cardiac cycle, given byPP_(n)=SAP_(n)−DAP_(n), with SAP_(n) and DAP_(n) being respectivelysystolic and diastolic arterial pressure in that cycle.

The pulsatile ABP waveform that results from simulating the model(EQ. 1) with P(0)=0 mmHg, T_(n)=1 s (such that heart rate in the n^(th)cycle HR_(n)=60 bpm), SV_(n)=100 ml, R=1 mmHg/(ml/s), and C=2 ml/mmHg,is shown in FIG. 3 b. The resulting steady-state pulse pressure equals50 mmHg.

Applicants define T_(n) to be the duration of the n^(th) cardiac cycle,i.e., the beat that begins at time t_(n) and ends at time t_(n+1) (soT_(n)=t_(n+1)−t_(n)). It follows that the average cardiac output in then^(th) cycle is given by:

$\begin{matrix}{{CO}_{n} = {\frac{{SV}_{n}}{T_{n}} = {C_{n}\frac{{PP}_{n}}{T_{n}}}}} & ( {{EQ}.\mspace{14mu} 4} )\end{matrix}$where the first equality is simply the definition and the second followson substituting from (EQ. 3).

Beat-to-Beat Averaged Windkessel Model

Given pulse pressure, (EQ. 4) may be used to estimate values of cardiacoutput. However, since the relation (EQ. 3) is based entirely on theessentially instantaneous ejection period assumed in this model,Applicants have recognized that the CO estimate obtained via (EQ. 4)does not take advantage of information from the remainder of the cardiaccycle that could be harnessed to provide a better-conditioned estimate.Specifically, the fact that (EQ. 1) interrelates the variables duringthe entire cardiac cycle, and indeed from one cycle to the next, has notbeen exploited in the derivation so far. Applicants recognize that tobetter reflect intra-cycle and inter-cycle behavior, one can average(EQ. 1) over an entire cardiac cycle rather than just the ejection phaseas follows:

$\begin{matrix}{{{\frac{C_{n}}{T_{n}}{\int_{t_{n}}^{t_{n + 1}}{\frac{\mathbb{d}{P(t)}}{\mathbb{d}t}\ {\mathbb{d}t}}}} + {\frac{1}{T_{n}R_{n}}{\int_{t_{n}}^{t_{n + 1}}{{P(t)}\ {\mathbb{d}t}}}}} = {\frac{1}{T_{n}}{\int_{t_{n}}^{t_{n + 1}}{{Q(t)}\ {{\mathbb{d}t}.}}}}} & ( {{EQ}.\mspace{14mu} 5} )\end{matrix}$where the time constant τ_(n)=R_(n)C_(n), and where we consider R_(n)and C_(n) to be constant within each cardiac cycle, but allow them tovary from cycle to cycle. Note that in this application, the words cycleand beat are used interchangeably. This application of cycle-to-cycle orbeat-to-beat averaging is an example of a general method known as themodulating function technique [17], first proposed by Shinbrot [18].This averaging yields the following relation over the n^(th) cycle [15]:

$\begin{matrix}{{{{C_{n}\frac{\Delta\; P_{n}}{T_{n}}} + \frac{\overset{\_}{P_{n}}}{R_{n}}} = {CO}_{n}}{where}} & ( {{EQ}.\mspace{14mu} 6} ) \\{{\Delta\; P_{n}} = {{P( t_{n + 1} )} - {P( t_{n} )}}} & ( {{EQ}.\mspace{14mu} 7} )\end{matrix}$is the beat-to-beat pressure change at the onset times, and

$\begin{matrix}{{\overset{\_}{P}}_{n} = {\frac{1}{T_{n}}{\int_{t_{n}}^{t_{n + 1}}{{P(t)}\ {\mathbb{d}t}}}}} & ( {{EQ}.\mspace{14mu} 8} )\end{matrix}$is the average or mean ABP computed over the n^(th) cycle. Note that(EQ. 6) is a natural discrete-time counterpart to (EQ. 1), with thefirst and second terms now representing average flow through thecapacitor and resistor, respectively, in the n^(th) cycle.

Combining (EQ. 4) and (EQ. 6), one can obtain

$\begin{matrix}{{{\frac{\Delta P}{T_{n}} + \frac{\overset{\_}{P_{n}}}{\tau_{n}}} = \frac{{PP}_{n}}{T_{n}}},} & ( {{EQ}.\mspace{14mu} 9} )\end{matrix}$where τ_(n)=R_(n)C_(n) is the only unknown. Applicants refer to themodel in (EQ. 9) as a beat-to-beat averaged Windkessel model. Note thatthis model has two elements and a source, and is thus a 2-elementbeat-to-beat averaged Windkessel model. Applicants note that thisderivation, and its application in estimation cardiac output may beextended to Windkessel models with 3 or more elements.

Because determination of central PP_(n) from peripheral pressurewaveforms is problematic due to wave reflections, in one embodiment,Applicants use an expression presented in [9] to estimate PP_(n) interms of the mean pressure P _(n) in the n^(th) cycle and DAP_(n):PP _(n)=α( P _(n) −DAP _(n))   (EQ. 10)where α is a constant. Assuming a triangular pulse shape yields α=2 (seeAppendix I), but Applicants' results on application of the CO estimationmethodology are not extremely sensitive to the precise value of α.

(EQ. 9) may be used to estimate 1/τ_(n) from knowledge of the remainingquantities, most of which are illustrated in FIG. 4. Applicants proceedto describe an estimation algorithm or scheme for estimating cardiacoutput based on the beat-to-beat averaged Windkessel model of (EQ. 9).

Estimation Using Least-Squares Error Criterion

Specifically, using a data window comprising an odd number of beatscentered at n, and assuming τ_(n) to be essentially constant over thiswindow, we invoke (EQ. 9) for each of the beats in the window to obtaina set of linear equations in the single unknown 1/τ_(n), as illustratedin Appendix II. The least-square-error solution of this set yields thedesired estimate. Repeating the process on a sliding window produces anestimate of 1/τ_(n) for every beat._Cardiac output can now be estimatedfrom (EQ. 6), rewritten below to show dependence on 1/τ_(n):

$\begin{matrix}{{CO}_{n} = {C_{n}( {\frac{\Delta\; P_{n}}{T_{n}} + \frac{\overset{\_}{P_{n}}}{\tau_{n}}} )}} & ( {{EQ}.\mspace{14mu} 11} )\end{matrix}$Note that the term

$( {\frac{\Delta\; P_{n}}{T_{n}} + \frac{\overset{\_}{P_{n}}}{\tau_{n}}} )$is the uncalibrated beat-by-beat cardiac output. The conventionalexpression for calibrated or uncalibrated CO_(n) neglects beat-to-beatvariability and therefore omits the term ΔP_(n)/T_(n); it is thusactually valid only in cyclic steady state, while (EQ. 11) holds moregenerally. In cyclic steady-state, the first term on the right hand sideof (EQ. 11) vanishes and the equation reduces to:

$\begin{matrix}{{CO}_{n} = \frac{\overset{\_}{P_{n}}}{R_{n}}} & ( {{EQ}.\mspace{14mu} 12} )\end{matrix}$which is simply the relation governing average flow through the resistorR_(n) given the mean pressure P_(n) . The vanishing term,

$\frac{\Delta\; P_{n}}{T_{n}}$in (EQ. 11), represents the average flow through C_(n). It is a measureof the beat-to-beat or inter-beat or inter-cycle variability in cardiacoutput and allows us to fine tune our CO estimate. The term

$\frac{\overset{\_}{P_{n}}}{R_{n}}$is a measure of the intra-cycle or intra-beat variability. Thedetermination of C_(n) using calibration information is discussed in thenext subsection.

Calibration of the Uncalibrated Beat-to-Beat Cardiac Output

To compute CO_(n) using (EQ. 11), one may estimate the compliance C_(n)by calibrating CO against one or more true or reference CO measurements.Note that in this application, true cardiac output is denoted TCO. Anatural calibration criterion is the root-mean-square-normalized error(RMSNE) at the calibration points, as described in Appendix V, i.e., thepoints at which true or reference cardiac output measurements areavailable. If the compliance C_(n) is assumed to be a constant equal toC, then it is straightforward to choose the C that minimizes the RMSNE[15]. Cohen et al. [10] instead used a mean calibration, dividing themean of the true CO values by the mean of the estimated CO values atthose points.

Considerably better results can be obtained by using a state-dependentmodel for C_(n), namely to model the lumped arterial compliance as afunction of arterial blood pressure. A simple choice is to assume anaffine dependence of C_(n) on mean arterial blood pressure, P_(n) , asfollows:C _(n)=γ₁+γ₂ P _(n)   (EQ. 15)

Though C may be expected to show a nonlinear dependence on mean arterialblood pressure, P, we assume here that P_(n) changes slowly enough thatC_(n) may be assumed essentially constant over any window of a few beatsor cycles in duration.

The calibration (EQ. 15) can be performed using a least-square-errorsolution to a linear system of equations as described in Appendix III.Other parameterizations may also be used, as discussed below and inAppendix III.

Estimation of Total Peripheral Resistance

Total peripheral resistance is also an important cardiovascularvariable. In the clinical setting, TPR is defined as the ratio of meanarterial blood pressure to cardiac output. However, taking into accountbeat-to-beat variability as in (EQ. 11) and thereby accounting fortransient flow into the arterial compliance, as was done by Toorop andco-workers [19], yields the modified expression for calibrated estimatedbeat-to-beat TPR:

$\begin{matrix}{R_{n} = \frac{\overset{\_}{P_{n}}}{{CO}_{n} - {C_{n}\frac{\Delta\; P_{n}}{T_{n}}}}} & ( {{EQ}.\mspace{14mu} 13} )\end{matrix}$In another embodiment, Applicants estimate beat-to-beat TPR using:

$\begin{matrix}{R_{n} = \frac{\tau_{n}}{C_{n}}} & ( {{EQ}.\mspace{14mu} 14} )\end{matrix}$Since both τ_(n) and C_(n) may be outputs of our estimation method, (EQ.14) may be easily implemented. In addition, if C_(n) is not available,uncalibrated beat-to-beat TPR may be estimated as R_(n)=τ_(n), or byusing the formula in (EQ 13) multiplied by C_(n), and taking (EQ. 9)into account.

TPR estimate (EQ. 14) is relatively smooth given that τ_(n) and C_(n)are estimated in a least-squares sense over a window of many beats.Since (EQ. 13) uses beat-to-beat variations in the flow to C_(n), ittends to be noisier than (EQ. 14). Nonetheless, for all the resultsdescribed later, Applicants used (EQ. 13) to estimate calibratedbeat-to-beat TPR.

Animal Data Set and Experiments

Applicants have tested their CO estimation method as outlined above onthe porcine dataset used by Cohen et al. [10]—a study on Yorkshire swineweighing 30-34 kg approved by the MIT Committee on Animal Care. Theanimals were intubated under anesthesia and mechanically ventilated.Once intubated, the animals' chests were opened, pressure and flowtransducers were placed, and over the course of 2-3 hours, CO, ABP, andHR were varied by one or more of the following interventions: volumeinfusions, slow hemorrhage, intravenous (IV) drugs (one or more ofphenylephrine, isoproterenol, esmolol, nitroglycerine, or dobutamine).FIG. 5 contains a table which gives a summary of population statisticsfor the six swine.

The data set contains measurements of ECG, central arterial bloodpressure (cABP), radial arterial blood pressure (rABP), femoral arterialblood pressure (fABP), and aortic flow (AF), all sampled at 250 Hz with16-bit amplitude resolution. The cABP waveform for swine 1 and the fABPwaveform for swine 4 were corrupted because the pressure transducermeasuring these variables were mis-calibrated during the experimentalprotocol. In addition, as was done in [10], data points at the end ofeach record, after progressive hemorrhage was started, were neglected asmeasured cardiac output was too low in these regions.

Using standard open-source algorithms [20], [21] on the AF waveform,Applicants derived onset times for each cardiac beat and HR. Applicantsalso calculated systolic and diastolic cABP, systolic and diastolicrABP, systolic and diastolic fABP, mean cABP, mean rABP, and mean fABPfor each swine. True or reference beat-to-beat CO was calculated byaveraging the AF waveform over each beat, and then applying a 50-beatmedian filter to the resulting output. All the data processing andcardiac output and TPR estimation algorithms were implemented in MATLAB™R14 (Mathworks Inc., Natick, Mass.).

Experimental Results and Discussion

Unless noted otherwise, the results reported for estimated cardiacoutput, or ECO, herein were generated using a 100-point state-dependent,i.e., mean pressure-dependent calibration to obtain C_(n) for eachanimal, as described in Appendix III. This amount of data representsless than 1% of each animal's data record, though the results changeminimally if as few as 10 or as many as 1,000 points are used tocalibrate. Also, all the results reported herein were obtained using theend-diastolic pressures in the expression for ΔP_(n). Applicants couldalso have used end-systolic pressures or pressures from another point inthe arterial blood pressure waveform, too.

FIG. 6 contains a table summarizing the error obtained for each animalusing either the cABP, rABP or fABP waveforms to estimate CO. Differentwindow sizes and values of α in (EQ. 10) yield similar errors as shownin Appendix IV. Applicants' results show mean RMSNEs of about 12%, whichis lower than the 15% reported in the literature as being acceptable forclinical purposes [13].

FIG. 9 shows the true and estimated CO, HR, mean rABP, true andestimated TPR, and drug infusions for Animal 1, FIG. 10 shows the trueand estimated CO, HR, mean fABP, true and estimated TPR, and druginfusions for Animal 2, and FIG. 11 shows the true and estimated CO, HR,mean cABP, true and estimated TPR, and drug infusions for Animal 3. Thespikes in the HR and R waveforms are a result of not filtering T_(n) orΔP_(n). In FIGS. 9, 10, and 11, Estimated CO and TPR track true CO andTPR very well while all major hemodynamic variables are variedindependently over a wide range. Furthermore, due to the continuousnature of our CO and TPR estimates, we track the effects ofpharmacological interventions quite closely.

FIG. 12 is a Bland-Altman plot for the CO estimation error using rABP.This plot is an aggregate of all 82,734 comparisons listed in the tablein FIG. 6. Note that ‘sp calib’ or ‘sp calibration’ denotes a 100-pointstate-dependent calibration, described in detail in Appendix III. Themean estimation error (or bias) is 18 ml/min, while the 1-standarddeviation (SD) of the estimation error is 429 ml/min. The flow probe(T206 with A-series attachment, Transonic Systems Inc., Ithaca, N.Y.)used in the animal experiments had a relative precision of ±5%, which atthe instrument scale of ±20 l/min is approximately 1 l/min. The 2-SDline for our estimate lies 860 ml/min from the line representing meanestimation error, showing that our method works very well when comparedto the flow probe measurements.

FIG. 13 shows a linear regression visualizing the CO estimation errorusing rABP. This plot is an aggregate of all 82,734 comparisons listedin the table in FIG. 6. Note that sp calib or sp calibration denotes a100-point state-dependent calibration, described in detail in AppendixIII. The regression coefficients for ECO versus TCO (with 95% confidenceintervals) are summarized in the table in FIG. 7. FIG. 8 contains atable which summarizes the correlation coefficients of the estimationerror versus mean pressure, heart rate, and TCO, with 95% confidencelevels. These correlation coefficients show that the estimation error isnot strongly correlated with mean ABP, HR, or TCO.

Comparison of Experiment Results to other Windkessel-based CO EstimationMethods

Applicants compared their animal experiment results to those obtainedusing the method of Cohen et al. and several Windkessel-based COestimation methods. Mukkamala et al. [10] (Mukkamala and Cohen areco-inventors on the Cohen et al. patent referenced herein) reported theresults shown in the table in FIG. 14, where Applicants calculated theaggregate RMSNEs using (EQ. 29). They re-sampled the 250 Hz data at 90Hz, and used a 6-minute window size, with a 3-minute overlap betweensuccessive windows. In each 6-minute window, Mukkamala et al. estimatedthe time constant τ from the impulse response of the Windkessel model,which they estimated by assuming the Windkessel model could berepresented by a 23-coefficient ARMA model with an impulsive pulsepressure source as its input [10]. In contrast, Applicants' model is a2-element beat-to-beat averaged Windkessel model which does not assume aparticular form of an impulsive input flow waveform.

Applicants' results are significantly different by swine and are, in anaggregate sense, much better than those reported in Mukkamala et al.[10]. In addition, instead of 6-minute windows, Applicants' method canuse much less data e.g. 10-50 beats to obtain an estimate of τ_(k). Acriticism of Mukkamala et al. put forward by van Lieshout et al. [22],and contended in [23], is that while CO estimates producedintermittently, e.g., every 3 minutes, may be good enough to trackslower patient dynamics, they may not be good enough for large, suddenchanges in CO, as is evident from patient data in the literature [24].Furthermore, Cohen and co-workers' estimate assumes a constant arterialtree compliance, which is not necessarily a valid assumption, but doesallow for the possibility of using only one true or reference CO pointfor calibration.

To do a fair comparison when comparing Applicants' estimated CO to theother Windkessel model-based estimates, Applicants used a meancalibration for each estimate. Applicants also calculated mean RMSNEsfor each estimate method without weighting the individual swine RMSNEsby the number of comparisons in each swine record. The results of thisanalysis are shown in the table in FIG. 15. Overall, the resultsobtained by applying Applicants' method (even without thestate-dependent calibration) and the methods proposed by Herd andMukkamala produce essentially equivalent results, and they outperformthe other methods tested. If Applicants were to apply the same 100-pointstate-dependent calibration on the Herd estimate, similar aggregateRMSNEs are obtained as in the table in FIG. 6 [15]. On the other hand,results obtained using human ICU data [25] show that the method ofLiljestrand and Zander [6] outperforms the Windkessel model-based COestimation methods we have used here [15]. In other published results onother human and animal data sets [15], Applicants' method performs muchbetter than the Cohen et al., Herd, and Liljestrand and Zander methods[15].

Pressure-Dependent Lumped Arterial Compliance

In the human cardiovascular system, arterial tree compliance is afunction of arterial blood pressure, and is perhaps better modeled assuch rather than as a constant. Furthermore, it is well-known thatarterial tree compliance depends on age (as we grow older, our arteriesget less elastic and arterial compliance decreases [26], [27]), gender,and disease state, e.g., arteriosclerosis results in lower compliance[28]. However, this compliance also depends on arterial blood pressure.

Applicants investigated the use of both a constant and astate-dependent, i.e., mean pressure-dependent compliance in calibratingthe uncalibrated beat-to-beat CO estimates. While there is muchdisagreement in the research community on this topic, CO estimationmethods exist which assume a constant arterial compliance forcalibration, while there are others which assume a pressure-dependentcompliance function—whether linear or nonlinear.

Some previous work on CO estimation suggests that the arterial treecompliance is constant over a wide range of mean arterial bloodpressures [8], [29], [30], [31]. However, there is no consensus on thisobservation. In fact, researchers have found that the calibration factorfor cardiac output, the equivalent of arterial tree compliance, can varysignificantly when estimating cardiac output (see FIG. 6 in [32], forexample) in humans. While constant compliance may have been observed inthe largest arteries in the body, it does not necessarily also hold truefor the smaller arteries, as shown by Liu et al. [33] and Cundick et al.[32]. In addition, researchers such as Burattini et al. [34], whoconducted canine experiments, have shown that arterial compliance canchange drastically in response to vasoactive drugs—partly due to theeffect of these drugs on the mean arterial blood pressure and partly dueto drug-induced changes in the mechanical properties of the arterialwall.

Other researchers have carefully investigated total arterial complianceand its dependence on mean arterial blood pressure. Westerhof andco-workers [35], [19], [36], [37], for example, have argued that thearterial tree volume depends strongly on pressure—falling sharply atlower pressures and asymptotically converging to a maximum at highpressures. The (incremental) compliance, therefore, is large at low meanpressures and steadily decreases with increasing pressure. In theirwork, they explored the use of such a nonlinear arterial treevolume-pressure function in various incarnations of the Windkesselmodel. Liu et al. [33] compared several nonlinear arterialvolume-pressure relationships, including logarithmic,piecewise-parabolic, and exponential relationships, and a specificlinear volume-pressure relationship—with corresponding constantcompliance. They argue that for the larger arteries, e.g., the aorticarch and thoracic aorta, a linear fit to the volume-pressure data wassufficient, but for the carotid, femoral, and brachial arteries, anonlinear relationship fit the volume-pressure data better. In [38],[39], the authors proposed several nonlinear arterial volume-pressurefunctions and evaluated them using simulated and human data.

Of particular relevance in terms of CO estimation, is the arctangentvolume-pressure curve proposed by Langewouters et al. [40] (note thatLangewouters and Wesseling are co-workers) based on ex vivo studies ofhuman thoracic and abdominal aortas. Their work was further strengthenedby the work of Tardy et al. [41] who describe in vivo studies on themechanical properties of human peripheral arteries. The relationshipproposed by Langewouters and co-workers, and used by Wesseling et al.[7] in a CO estimation method, yields the following (incremental)arterial compliance, C_(a):

$\begin{matrix}{C_{a} = \frac{\alpha_{1}}{\alpha_{2} + {\alpha_{3}( {V_{a} - V^{*}} )}^{2}}} & ( {{EQ}.\mspace{14mu} 16} )\end{matrix}$where α₁, α₂, and α₃ are constants, and V* is the inflection point oftheir arctangent aortic volume-pressure relationship. In humans, a valueV*=40 mmHg is suggested [7]. The constants α₁, α₂, and α₃ depend onpatient gender and age; nominal values of these constants can beextracted from regression analyses described in [40]. In contrast,Applicant's proposed calibration factor (EQ. 15) has only twoparameters. Furthermore, it does not depend nonlinearly on pressure,making it much more amenable to a linear least-squares solution asdescribed in Appendix III.

The CO estimation approach in [7] allows a further adjustment of α₁ whencalibrating against available CO measurements. There are otherpressure-dependent compliances that have been used in CO estimationmethods, e.g., the pressure-dependent compliances of Godje et al. [42]and Liljestrand and Zander [6]. In [6], compliance is simply modeled asbeing inversely proportional to the sum of the beat-to-beat systolic anddiastolic arterial blood pressures, while in [42], compliance is modeledusing a complicated expression that involves both mean and instantaneousarterial blood pressure.

In Applicants' own work, Applicants attempt to use either a lineararterial tree compliance as given by (EQ. 15) above, or a constantarterial tree compliance:C_(n)=γ₁   (EQ. 17)which arises naturally from its linear counterpart (15) as the specialcase of γ₂=0.

The function (EQ. 15) corresponds to a parabolic volume-mean pressurerelationship in the arterial tree, is simpler than the one used in [7],and facilitates estimation of patient- or animal-specific parametersfrom calibration data. A review of the literature shows no significantadvantages of a logarithmic or arctangent volume-mean pressurerelationship over one that is parabolic or one that uses instantaneousarterial blood pressure.

To test the hypothesis that a state-dependent compliance may be moreappropriate than a constant compliance, Applicants applied an embodimentof their CO estimation method and other methods from the literature tothe porcine data set using both a 100-point state-dependent calibration,and a 100-point mean calibration. These results may not berepresentative of other animal and/or human data sets as discussed in[15]. In each swine record, the points were spread evenly throughouteach swine record. The results of this experiment are shown in the tablein FIG. 16. It is clear that a state-dependent calibration, even on just100 points out of 10,000-15,000 yields better results than a meancalibration. Applicants analyzed their linear mean pressure-dependentarterial tree compliance for each of the six swine and discovered that,apart from Swines 1, 4, and 6, all the swines have an almost-constant,i.e., pressure-independent, arterial compliance, except for certainsections of the data. FIGS. 17 and 18( a) show a time series of our fitfor the arterial compliance, and a plot of true CO estimated CO and ourfit for the arterial compliance versus mean pressure, respectively, forswine 1. Note that in FIG. 18, we used 100 points spread evenlythroughout each swine record to compute our fit for the arterialcompliance C. Graphs of true CO/estimated CO and our fits for thearterial compliance versus mean pressure for all six swine appear inFIG. 18.

It is clear from FIGS. 17 and 18, that for the majority of the datapoints in each swine record, the arterial tree compliance is essentiallyconstant. Thus, for all but two of the swines in this data set—swines 1and 4—we can assume that γ₂ in (EQ. 15) is zero, except for certainsections of the data. Thus, for all but swine 4, there are sets of datapoints, particularly at lower mean pressures, which would be wellcaptured by a linear compliance instead of a constant compliance.

Inter-Beat or Beat-to-Beat Variability

The table in FIG. 16 seems to imply that the Herd CO estimation methodand Applicants CO estimation method are, in an aggregate sense,equivalent. However, the same is not true if Applicants take a closerlook at the true and estimated CO waveforms, particularly in sectionswhere inter-beat variability is high.

Applicants CO estimation method incorporates, among other things,beat-to-beat variability and therefore may produce more accurate COestimates than those produced by many intra-beat CO estimation methods.To test this hypothesis, i.e., that beat-to-beat variability improvesour CO estimate, Applicants define a beat-to-beat variability index,B2BVI_(b) (%), in each 360-beat window as follows:

$\begin{matrix}{{B\; 2\; B\; V\; I_{b}} = {\frac{1}{360}{\sum\limits_{n = b}^{b + 360}\;{( {100\frac{\Delta\; P_{n}}{{PP}_{n}}} ).}}}} & ( {{EQ}.\mspace{14mu} 18} )\end{matrix}$

Applicants calculated RMSNEs only using points on the estimated COwaveform where τ_(n) was calculated on windows where B2BVI_(b)≧5%. Theresults obtained are summarized in the tables of FIGS. 19 and 20, whereApplicants' estimate is compared to the results obtained using the Herdestimation method. For some animals, there were no such windows asdepicted with a ‘-’ in the table. It is clear that on windows with highbeat-to-beat variability, Applicants' CO estimates performs eithercomparably (in the case of swine 2) or a lot better than the Herdestimate. This result is independent of whether a mean or astate-dependent calibration is performed, as seen in FIGS. 19 and 20.Thus, in data segments in which beat-to-beat variability is significant,as reflected by the ratio ΔP_(n)PP_(n) in (EQ. 18), Applicants' methoddoes substantially better than static pulse contour methods, such as theHerd method, that solely analyze the intra-beat pulse shape.

Appendix I: Derivation for a in (EQ. 10)

In an embodiment, if one assumes a high enough HR (i.e. T_(n)<<τ_(n)) inthe n^(th) cardiac cycle of the Windkessel model (EQ. 1), one can seethat in the (n+1)^(st) cardiac cycle, diastolic ABP is given by

$\begin{matrix}{{DAP}_{n + 1} = {{{SAP}_{n}e^{-}\frac{T_{n}}{\tau_{n}}} \approx {{SAP}_{n}\lbrack {1 - \frac{T_{n}}{\tau_{n}}} \rbrack}}} & ( {{EQ}.\mspace{14mu} 19} )\end{matrix}$such that the mean ABP in the n^(th) cardiac cycle may be approximatedas

$\begin{matrix}{\overset{\_}{P_{n}} \approx {\frac{1}{T_{n}}\lbrack {{{DAP}_{n + 1}T_{n}} + {\frac{1}{2}{T_{n}( {{SAP}_{n} - {DAP}_{n}} )}}} \rbrack}} & ( {{EQ}.\mspace{14mu} 20} )\end{matrix}$which yields the following formula for pulse pressure in the n^(th)cardiac cycle:PP _(n) =SAP _(n) −DAP _(n)≈2( P _(n) −DAP_(n)).   (EQ. 21)

Appendix II: More on Linear Least-Squares Estimation for 1/τ_(n)

In an embodiment described above, Applicants assumed that 1/τ_(n) variesslowly from beat-to-beat and that it stays fixed over several beats. HadApplicants not assumed an impulsive cardiac ejection in (EQ. 3), therewould have been two unknowns in (EQ. 9), and Applicants would have hadto make the assumption that both 1/τ_(n) and

$\frac{{SV}_{n}}{C_{n}}$vary slowly from beat-to-beat. Such an assumption may be invalid forstroke volume as it can change rapidly from one beat to the next. Inaddition, depending on the data set used, the resulting two-parameterleast-squares estimation problem may be ill-conditioned. However, insome embodiments, the resulting two-parameter least-squares estimationproblem may be feasible and well-conditioned.

Applicants estimated CO directly from (EQ. 9) by computing aleast-squares estimate of 1/τ_(n) over a data window, i.e., Applicantscalculated a least-squares estimate of 1/τ_(n) for the n^(th) beat usinga window comprising the k/2 adjacent beats on each side of this beat.This results in a total of k (even) equations in one unknown, awell-conditioned least-squares estimation problem as shown immediatelybelow:

$\begin{matrix}{{\begin{bmatrix}{- {\overset{\_}{P}}_{n - \frac{k}{2}}} \\\vdots \\{- {\overset{\_}{P}}_{n + \frac{k}{2}}}\end{bmatrix}\lbrack \frac{1}{\tau_{n}} \rbrack} = \begin{bmatrix}{\frac{\Delta\; P_{n - \frac{k}{2}}}{T_{n - \frac{k}{2}}} - \frac{{PP}_{n - \frac{k}{2}}}{T_{n - \frac{k}{2}}}} \\\vdots \\{\frac{\Delta\; P_{n + \frac{k}{2}}}{T_{n + \frac{k}{2}}} - \frac{{PP}_{n + \frac{k}{2}}}{T_{n + \frac{k}{2}}}}\end{bmatrix}} & ( {{EQ}.\mspace{14mu} 22} )\end{matrix}$In one embodiment, Applicants assign the estimate 1/τ_(n) from eachwindow to the midpoint of that window, and set n>k/2 in (EQ 22).

Appendix III: More on Calibration Methods

In calibrating the uncalibrated beat-to-beat cardiac output estimates,one may attempt to find a value for C_(n) such that the CO estimationerror, ε_(n), inCO _(n) =C _(n) UCO _(n)+ε_(n)   (EQ.23)is minimized, in some sense, for all n of interest. Note that in (EQ.23), we define uncalibrated cardiac output, or UCO, by:

$\begin{matrix}{{U\; C\; O_{n}} = ( {\frac{\Delta\; P_{n}}{T_{n}} + \frac{\overset{\_}{P_{n}}}{\tau_{n}}} )} & ( {{EQ}.\mspace{14mu} 24} )\end{matrix}$

For example, one can find C_(n) such that theroot-mean-square-normalized-error (RMSNE), described in the Appendix V,is minimized, i.e., find the optimal C_(n) such that

$\frac{\in_{n}}{C\; O_{n}}$in

$\begin{matrix}{1 = {\frac{C_{n}U\; C\; O_{n}}{C\; O_{n}} + \frac{\in_{n}}{C\; O_{n}}}} & ( {{EQ}.\mspace{14mu} 25} )\end{matrix}$is minimized, in the least-squares sense, for all k of interest. Notethat this kind of least-squares calibration may be done because of thesimple form for C_(n) in (EQ. 15).

In another embodiment, one may proceed as follows: given a set of COmeasurements at points {p₁, . . . ,p_(n),}, {CO_(pi)}, find theleast-squares optimal γ₁ and γ₂ in (EQ. 15), by solving (EQ. 25) usingat least two reference or true CO (TCO) measurements:

$\begin{matrix}{{\begin{bmatrix}\frac{U\; C\; O_{p_{1}}}{C\; O_{p_{1}}} & {{\overset{\_}{P}}_{p_{1}}\frac{U\; C\; O_{p_{1}}}{C\; O_{p_{1}}}} \\\vdots & \vdots \\\frac{U\; C\; O_{p_{n}}}{C\; O_{p_{n}}} & {{\overset{\_}{P}}_{p_{n}}\frac{U\; C\; O_{p_{n}}}{C\; O_{p_{n}}}}\end{bmatrix}\begin{bmatrix}\gamma_{1} \\\gamma_{2}\end{bmatrix}} = \begin{bmatrix}1 \\\vdots \\1\end{bmatrix}} & ( {{EQ}.\mspace{14mu} 26} )\end{matrix}$for n≧2. For (EQ. 26) to be well-conditioned, there should be enoughvariation in mean pressure P_(k) . If we solve (EQ. 26) using at leasttwo equally-spaced true CO measurements, Applicants call this atwo-point state-dependent calibration. In the results described in thisapplication, a 100-point state-dependent calibration was used. Astate-dependent calibration may be more realistic in settings such asthe intensive care unit (ICU) where CO is measured only intermittently.

The results reported in Mukkamala et al. [10] were generated using aC_(n) that is not optimal in the sense of (EQ. 25). In Mukkamala et al.[10], a mean calibration was done such that:

$\begin{matrix}{C_{n} = {C = {\frac{{mean}(C)}{{mean}( {U\; C\; O} )}.}}} & ( {{EQ}.\mspace{14mu} 27} )\end{matrix}$This calibration is equivalent to assuming that γ₂ in (EQ. 15) equals 0and that γ₁ is given by (EQ. 27) above.

Appendix IV: Results for Different Data Window Sizes and Values of α

Applicants used various window sizes (i.e. number of beats)—roughlyranging from 6 seconds to 12 minutes of data—to estimate τ_(n), andhence CO_(n). Applicants observed that mean RMSNEs do not changesignificantly for window sizes above 50 beats, implying that one doesnot seem to need variability beyond the range of a 50 beats (or 30seconds at a resting porcine heart rate of 100 bpm) to obtain reasonablecalibrated beat-to-beat CO estimates. This observation, however, couldbe strongly dependent on the porcine data set used.

In various embodiments, Applicants have used various values for α in(EQ. 10) to estimate CO_(n). With a window size equal to 360 beats, themean RMSNE taken over the six swine for each value of a were about thesame for a ranging from 1.5 to 100. For small α e.g. α≈0.01-0.9, themean RMSNEs are much higher than with α≧1.5. For other window sizes, thesame result holds i.e. the mean RMSNEs are not too sensitive to thevalue of α except for small α. From a least-squares estimation point ofview, this is not surprising as the constant α must be large enough thatthe term

$\frac{1}{T_{n}}{PP}_{n}$in (EQ. 9) is of the same order of magnitude as, i.e. significant, asthe term

$- {\frac{\overset{\_}{P_{n}}}{\tau_{n}}.}$

Appendix V: Root Mean Square Normalized Error

In comparing true cardiac output to estimated cardiac output (ECO),Applicants used a root-mean-square-normalized-error criterion. For aparticular swine, e.g. swine i, given n_(i) points at which true CO wasmeasured and CO was estimated, the RMSNE (in %) for the CO estimate forswine i, denoted RMSNE_(swinei), is given by the following formula:

$\begin{matrix}{{R\; M\; S\; N\; E_{{swine}_{i}}} = {\sqrt{\frac{1}{n_{i}}{\sum\limits_{n = 1}^{n_{i}}\;( \frac{100( {{trueCO}_{n} - {estimatedCO}_{n}} )}{{trueCO}_{n}} )^{2}}}.}} & ( {{EQ}.\mspace{14mu} 28} )\end{matrix}$

As the swine data records were of varying lengths, the “aggregate” RMSNEover all the swines was calculated as the weighted average of theindividual swine RMSNEs. Assuming that Σ_(i)n_(i)=N, the RMSNE over allswines is given by:

$\begin{matrix}{{R\; M\; S\; N\; E} = {\sqrt{\frac{1}{N}{\sum\limits_{i}\;{n_{i}( {R\; M\; S\; N\; E_{{swine}_{\cdot i \cdot}^{2}}} )}}}.}} & ( {{EQ}.\mspace{14mu} 29} )\end{matrix}$

Note that RMSNE is an aggregate measure of performance. While itrepresents how the true CO and estimated CO compare in an average sense,it does not classify the CO estimation error with regard to theparticular values of CO, ABP, or HR, or even the particularinterventions being performed on the animals. A linear regression oftrue CO versus estimated CO with a reported correlation coefficient mayalso only be an aggregate measure of performance, as would aBland-Altman (see [43], [44]) plot of CO error versus the mean of trueCO and estimated CO.

The invention may be embodied in other specific forms without departingfrom the spirit or essential characteristics thereof. The forgoingembodiments are therefore to be considered in all respects illustrative,rather than limiting of the invention.

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1. A method for estimating beat-by-beat cardiovascular parameters andvariables, comprising: processing two or more cycles of arterial bloodpressure to determine a mean arterial blood pressure over each cyclebeginning at a beat onset time and an inter-beat variability in bloodpressure between the two or more cycles; and computing estimates of oneor more cardiovascular parameters and variables from the mean arterialblood pressures, the inter-beat variability, and a beat-to-beat averagedWindkessel model of an arterial tree.
 2. The method of claim 1, whereinthe one or more cardiovascular parameters and variables include abeat-by-beat time constant of the arterial tree.
 3. The method of claim2, wherein the time constant is estimated over a data window.
 4. Themethod of claim 2, wherein the time constant is estimated throughoptimization of an error criterion.
 5. The method of claim 4, whereinthe error criterion is least-squared error.
 6. The method of claim 1,wherein the one or more cardiovascular parameters and variables includean uncalibrated beat-by-beat cardiac output.
 7. The method of claim 6,further comprising computing calibrated beat-by-beat cardiac output fromthe uncalibrated beat-by-beat cardiac output using a calibration factor.8. The method of claim 7, wherein the calibration factor is computed foreach of the cycles.
 9. The method of claim 7, wherein the calibrationfactor represents a lumped arterial compliance.
 10. The method of claim9, wherein the lumped arterial compliance is modeled as a function ofmean arterial blood pressure.
 11. The method of claim 9, wherein thelumped arterial compliance is modeled as a parameterized function ofmean arterial blood pressure.
 12. The method of claim 9, wherein thelumped arterial compliance is modeled as a two-parameter function ofmean arterial blood pressure.
 13. The method of claim 9, wherein thelumped arterial compliance is modeled as a constant.
 14. The method ofclaim 11, 12 or 13, wherein a parameter of the lumped arterialcompliance is estimated through optimization of an error criterion. 15.The method of claim 14, wherein the error criterion is least-squarederror.
 16. The method of claim 6, wherein the one or more cardiovascularparameters and variables include an uncalibrated beat-by-beat totalperipheral resistance.
 17. The method of claim 16, further comprisingcomputing calibrated beat-by-beat total peripheral resistance from aratio of the time constant to a lumped arterial compliance.
 18. Themethod of claim 16, further comprising computing calibrated beat-by-beattotal peripheral resistance from a ratio of mean arterial blood pressureto calibrated cardiac output.
 19. The method of claim 16, furthercomprising computing calibrated beat-by-beat total peripheral resistancefrom a ratio of mean pressure to a systemic blood flow.
 20. The methodof claim 19, wherein the systemic blood flow is calculated as thecalibrated cardiac output minus the product of lumped arterialcompliance with a ratio of beat-to-beat arterial blood pressure changeto beat duration.
 21. The method of claim 1, wherein the arterial bloodpressure is measured at a central artery of a cardiovascular system. 22.The method of claim 1, wherein the arterial blood pressure is measuredat a peripheral artery of a cardiovascular system.
 23. The method ofclaim 1, wherein the arterial blood pressure is measured using anoninvasive blood pressure device.
 24. The method of claim 23, whereinthe arterial blood pressure is measured using a photoplethysmographicblood pressure device.
 25. The method of claim 23, wherein the arterialblood pressure is measured using a tonometric blood pressure device. 26.The method of claim 1, wherein processing the two or more cycles ofarterial blood pressure includes obtaining a mean blood pressure, adiastolic blood pressure, and a systolic blood pressure for each cycle.27. The method of claim 1, wherein processing the two or more cycles ofarterial blood pressure includes obtaining the beat onset time for eachcycle.
 28. The method of claim 27, wherein processing the two or morecycles of arterial blood pressure includes computing a beat-to-beatarterial blood pressure change between consecutive beat onset times. 29.The method of claim 28, wherein processing the two or more cycles ofarterial blood pressure includes estimating pulse pressure in each cycleas a proportionality constant multiplied by a difference between meanpressure and diastolic pressure in each cycle.
 30. The method of claim29, wherein the proportionality constant in each cycle is fixed.
 31. Themethod of claim 1, wherein processing the two or more cycles of arterialblood pressure includes obtaining a beat duration for each cycle.
 32. Asystem for estimating beat-to-beat cardiac output comprising: a bloodpressure measuring device; a processor; a display; a user interface; anda memory storing computer executable instructions, which when executedby the processor cause the processor to: receive two or more cycles ofarterial blood pressure from the blood pressure device; analyze the twoor more cycles of arterial blood pressure to determine a mean arterialblood pressure over each cycle beginning at a beat onset time and aninter-beat variability in blood pressure between the two or more cycles;compute estimates of one or more cardiovascular system parameters andvariables from the mean arterial blood pressures, the inter-beatvariability, and a beat-to-beat averaged Windkessel model of an arterialtree; and display the estimates.
 33. The system of claim 32, wherein theblood pressure measuring device is a noninvasive blood pressuremeasuring device.
 34. The system of claim 33, wherein the noninvasiveblood pressure measuring device is a photoplethysmographic bloodpressure measuring device.
 35. The system of claim 32, wherein thenoninvasive blood pressure measuring device is a tonometric bloodpressure measuring device.
 36. The system of claim 32, wherein thearterial blood pressure is measured at a central artery of acardiovascular system.
 37. The system of claim 32, wherein the arterialblood pressure is measured at a peripheral artery of a cardiovascularsystem.
 38. The system of claim 32, wherein the one or morecardiovascular parameters and variables include a beat-by-beat timeconstant of the arterial tree.
 39. The system of claim 38, wherein thetime constant is estimated over a data window.
 40. The system of claim38, wherein the time constant is estimated through optimization of anerror criterion.
 41. The system of claim 39, wherein the error criterionis least-squared error.
 42. The system of claim 32, wherein the one ormore cardiovascular parameters and variables include an uncalibratedbeat-by-beat cardiac output.
 43. The system of claim 32, furthercomprising computer executable instructions which, when executed by theprocessor, cause the processor to compute calibrated beat-by-beatcardiac output from the uncalibrated beat-by-beat cardiac output using acalibration factor.
 44. The system of claim 43, further comprisingcomputer executable instructions which, when executed by the processor,cause the processor to compute the calibration factor for each of thecycles.
 45. The system of claim 43, wherein the calibration factorrepresents a lumped arterial compliance.
 46. The system of claim 45,wherein the lumped arterial compliance is modeled as a function of meanarterial blood pressure.
 47. The system of claim 45, wherein the lumpedarterial compliance is modeled as a parameterized function of meanarterial blood pressure.
 48. The system of claim 45, wherein the lumpedarterial compliance is modeled as a two-parameter function of meanarterial blood pressure.
 49. The system of claim 45, wherein the lumpedarterial compliance is modeled as a constant.
 50. The system of claim47, 48, or 49, wherein a parameter of the lumped arterial compliance isestimated through optimization of an error criterion.
 51. The system ofclaim 50, wherein the error criterion is least-squared error.
 52. Thesystem of claim 32, wherein the one or more cardiovascular parametersand variables include an uncalibrated beat-by-beat total peripheralresistance.
 53. The system of claim 52, further comprising computerexecutable instructions which, when executed by processor, cause theprocessor to compute calibrated beat-by-beat total peripheral resistancefrom a ratio of the time constant to a lumped arterial compliance. 54.The system of claim 52, further comprising computer executableinstructions which, when executed by the processor, cause the processorto compute calibrated beat-by-beat total peripheral resistance from aratio of mean arterial blood pressure to calibrated cardiac output. 55.The system of claim 52, further comprising computer executableinstructions which, when executed by the processor, cause the processorto compute calibrated beat-by-beat total peripheral resistance from aratio of mean pressure to a systemic blood flow.
 56. The system of claim55, wherein the systemic blood flow is calibrated cardiac output minusthe product of lumped arterial compliance with a ratio of beat-to-beatarterial blood pressure change to beat duration.
 57. The system of claim32, wherein analyzing the two or more cycles of arterial blood pressureincludes obtaining a mean blood pressure, a diastolic blood pressure,and a systolic blood pressure for each cycle.
 58. The system of claim32, wherein analyzing the two or more cycles of arterial blood pressureincludes obtaining the beat onset time for each cycle.
 59. The system ofclaim 58, wherein analyzing the two or more cycles of arterial bloodpressure includes computing a beat-to-beat arterial blood pressurechange between consecutive beat onset times.
 60. The system of claim 58,wherein analyzing the two or more cycles of arterial blood pressureincludes estimating pulse pressure in each cycle as a proportionalityconstant multiplied by a difference between mean pressure and diastolicpressure in each cycle.
 61. The system of claim 60, wherein theproportionality constant in each cycle is fixed.
 62. The system of claim32, wherein analyzing the two or more cycles of arterial blood pressureincludes obtaining a beat duration for each cycle.